Here \((a)\)  is due to the total probability law and \((b)\)  is due to the independence of \(\Phi _{(0),\mathrm {d}}\)  and the rest of the cluster process \(\Phi \) . Now, from the Slivnyak theorem [1]}, we know that \(\mathbb {E}^{!\mathbf {x}}[\ ]=\mathbb {E}[\ ]\) . Hence,
\({1}{-1}F_{R_\mathrm {N}}(r)&=1-\mathbb {E}_{\mathbf {x}_0}\left[\mathbb {E}^0{\left[\prod _{\mathbf {y}_{(i),j} \in \Psi }\hspace{-7.22743pt}\mathbb {1}\left({\Vert \mathbf {x}_0+\mathbf {y}_{(i),j}\Vert >r}\right)\right]}\right] \mathbb {E}\left[\prod _{\mathbf {x}_i}g(\mathbf {x}_i)\right].\) 
